IMC 2004

Let $A$ be a real $4\times 2$ matrix and $B$ be a real $2\times 4$ matrix such that $$AB = \left(\begin{array}{cccc} 1 & 0 & -1 & 0 0 & 1 & 0 & -1 -1 & 0 & 1 & 0 0 & -1 & 0 & 1 \end{array}\right).$$ Find $BA$ .

Let $D$ be the closed unit disk in the plane, and let $z_1,z_2,\ldots,z_n$ be fixed points in $D$ . Prove that there exists a point $z$ in $D$ such that the sum of the distances from $z$ to each of the $n$ points is greater or equal than $n$ .